Many mathematical problems involve analyzing data to determine relationships between variables. For example, in regression analysis an expression can be determined to describe data (which is sometimes referred to as ‘fitting’ the expression to the data). This is shown in FIG. 1A, which presents a drawing 100 illustrating the fitting a line to data. The equation for a line y (the independent variable) can be expressed asy=mx+b, where x (the data) is the dependent variable, and m and b are unknown coefficients (the slope and y-intercept, respectively) that are to be determined during the fitting. In this example, each datum in the data corresponds to a point in the x-y plane (such as x0, y0).
Typically, the minimum number of data points needed to uniquely determine the fitting equation equals the number of unknowns in the fitting equation (as shown in FIG. 1A, for a line, the minimum number of data points is two). If there are more data points than this minimum number, statistical techniques such as least-squares regression may be used to determine the unknown coefficients. However, if there are fewer data points available than the minimum number, it is typically not possible to uniquely determine the unknowns. This is shown in FIG. 1B, which presents a drawing 150 illustrating the fitting of multiple lines to a datum. In principle, there are an infinite number of equivalent fitting solutions that can be determined. This type of problem is sometimes referred to as ‘sparse’ or ‘underdetermined.’
Unfortunately, many interesting problems are underdetermined. For example, in biology, there are important differences in the central nervous systems within a species (i.e., from individual to individual) and between species. These differences in the central nervous system reflect underlying variations in the neurons and the neural circuits that contain them. Collectively, the neurons and their interconnections in an organism constitute its neurome, and these neurons can be identified and represented by their properties.
One approach for characterizing the properties of the neurons in an organism is to measure electrical signals associated with the activity of the neurons using electrodes. In principle, the measured electrical signals include valuable information about the organism's central nervous system. In practice, there are often an enormous number of electrical signals and a much smaller number of observations (such repeated cycles of a particular activity). As a consequence, the problem is underdetermined, and it can be difficult to identify a subset of the electrical signals that are relevant to the particular activity without an excessive number of false positives. These false positives can significantly increase the time and expense needed to analyze the electrical signals to identify the subset.
Therefore, there is a need for an analysis technique to identify associations in underdetermined problems without the problems listed above.